Degeneracy in One Dimensional Quantum Systems: Dynamically

Degeneracy in One Dimensional Quantum Systems: Dynamically Shifted Oscillator

Pirooz Mohazzabi1*, G. Clark Alexander2 1 Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI 53141, USA. 2 Department of Mathematical Sciences, DePaul University, Chicago, IL 60604, USA. * Corresponding author. Tel.: (262) 595-2529; email: [email protected] Manuscript submitted June 2, 2016; accepted November 2, 2016. doi: 10.17706/ijapm.2017.7.2.79-86

Abstract: We study the dynamically shifted quantum oscillator in one dimension, whose detailed

examination reveals some very interesting and unexpected results. While several articles have been written

on the classical dynamically shifted oscillator, the quantum version is somewhat more difficult since it

defies solution by standard algebraic and analytic techniques. In this work, we show that the analytic

approach to this problem results in a trivial, non-normalizable, and thus nonphysical solution. Therefore,

we undertake a numerical approach and find two strikingly different results. First, the energy eigenvalues

rise when the dynamic shift is positive, which is almost intuitive. The second result is surprising as the

system shows degeneracy in one dimension despite the fact that this potential violates two of the three

required criteria described previously by other investigators. This work, therefore, opens a door to finding

multiple classes of one-dimensional degenerate systems with continuous, symmetric potentials.

Key words: Degeneracy, one dimension, quantum, dynamically shifted, oscillator.

1. Introduction

The quantum dynamically shifted oscillator in one dimension is given by the potential

V x

k x x x

k x x x

( )

( ) ,

( ) ,

1

20

1

20

0

2

0

2

(1)

which reduces to the simple harmonic oscillator for x0=0. For x0>0 and x0<0 the potential is continuous but

not smooth at the origin, as shown in Fig. 1. In this article we show that several unusual phenomena occur

when x0 is small negative. For x0=0 the solution is well known and perhaps the most widely studied example

in one-dimensional quantum mechanics, while the case of x0>0 simply excites the ground state. The unusual

phenomena that occur when x0 is small negative are two-fold. First, the eigenvalues fall below the ground

state value of ½ ℏω as if the dynamic shift locally produces an anharmonic shift similar to the linear case, λx.

Second, when the reduced dynamic shift ξ0 becomes smaller than approximately -3, we begin seeing

degeneracy in eigenvalues producing two solutions of opposite parity.

The question of degeneracy in one dimension is interesting as most of the literature explores higher

dimensional degeneracy. In their paper, Kar and Parwani [1] explore the possibility of degenerate bound

states in one dimensional systems. Along with Koley and Kar [2], they present the following bottomless

International Journal of Applied Physics and Mathematics

79 Volume 7, Number 2, April 2017

potential,

V x A x A x( ) ( cosh sec ) 1

2

2

2 h (2)

with v>0 and A2 2 21

( ) , as giving two eigenstates of different parity,

1

1

2

1

1

2

( )cos( cosh )

cosh

( )sin( cosh )

cosh

/

/

xA xdx

x

xA xdx

x

(3)

These potentials, however, do not appear in elementary quantum systems, but brane scenarios. The

potential that we investigate, however, is very similar to that experienced by, for example, the Nitrogen atom

in an Ammonia molecule [3].

Fig. 1. Dimensionless plots of the dynamically shifted oscillator for positive and negative dynamic shifts.

2. The Mathematics of the Dynamically Shifted Oscillator

The standard method for solving a piecewise potential analytically is to solve each piece separately and

then match the solutions. In particular this technique has been applied not only in textbooks, but in

numerous articles involving oscillators with altered potentials, for example the biharmonic oscillator [4]

and the harmonic plus a delta-potential [5].

By the postulates of quantum mechanics, wave function solutions to equations with continuous potentials

are required to be continuous and have continuous first and second derivatives. Every derivative which



exists in the potential energy function requires another continuous derivative in the wave function solution.

Therefore, in the case of the dynamically shifted oscillator we must have at least two continuous derivatives.

The problem one experiences with trying to find a global solution is that the solutions to each half are well

known as the harmonic oscillator solutions. When x0 ≠ 0 one finds that matching solutions yields 0=0 which

is, of course, true but not useful.

To be explicit about this solution, let us define the following unitless quantities

mx

mx

E

1 2

0

1 2

0

/ /

, , (4)

The reduced time-independent Schrödinger equation has the solutions

n

n

n nx A H n( ) ( ) exp( )

( ), , , ,

1

20 1 2

0

2

0 (5)

where Hn are Hermite polynomials and An are constants. The upper and lower signs hold for ξ≥0 and ξ<0,

respectively, and the corresponding eigenvalues for each branch are

n n n 1

20 1 2, , , , (6)

Now matching each side of the solution we see that the following equations must be satisfied

A Hk k2 2 0 0

2 0( ) sinh( ) (7)

A Hk k2 1 2 1 0 0

2 0 ( ) cosh( ) (8)

These equations, however, are satisfied if and only if An=0 for all n unless ξ0=0. This clearly implies that no

global analytic eigenfunction can exist except for ξ0=0 (which is the case of the harmonic oscillator). The

way in which one may guarantee global analytic solutions is to have a smooth potential. It is, of course,

possible to have a smooth solution via matching solutions, but there is no guarantee that these will give

analytic solutions. The only mathematical guarantee is to have a smooth potential.

If one were to attempt to solve this algebraically, the standard approach of factoring the differential

operator in raising and lowering operators works exactly as expected. Essentially, there is no detectable

difference from the standard textbook treatment of the harmonic oscillator aside from having shifted the

minimum of the potential. The algebraical analysis predicts that we should recover the same eigenvalues as

the unshifted oscillator, which we can falsify via numerical techniques given in the succeeding sections.

The main problem in the solution of the dynamically shifted oscillator is that the analytic techniques for

solution do not account for the (unitless) eigenvalues dipping below the ground state value ε0=1/2. In fact,

none of the algebraic or analytic techniques show any sign of eigenvalues being dependent upon the value

of the dynamic shift. Numerically, however, this is what happens. The problem here is that we are dealing

with a system which appears innocuous, but Hartmann [6] and Mohazzabi [7] have shown that this has an

intrinsically different behavior.

3. Degeneracy Conditions in One Dimension

Kar and Parwani [1] explore degenerate bound states in one dimensional quantum mechanics with the

additional restriction that a potential be non-singular in any finite domain. Of course, the last stipulation is

what makes the question difficult. It is rather easy to simply place two infinite potential wells some distance

apart on the real line and solve the Schrödinger equation in each part then flip the sign of one of the



solutions. Moreover, many systems with an infinite singularity in a finite domain exhibit this kind of

degeneracy. The paper of Kar and Parwani, however, asks a much more interesting question. The conclusion

they draw is that a one-dimensional quantum system cannot be degenerate unless the following criteria are

met:

1) V(x) is a continuous function.

2) V(x) is unbounded from below at .

3) The eigenfunction has infinite derivative as x → .

The third criterion emphasizes that the momentum is divergent, which corresponds to divergent kinetic

energy. The only way to maintain a finite total energy is to have unbounded negative potential energy. While

the constructions given by Kar and Parwani [1] maintain all these criteria, we will show that there is yet

another degenerate bound state system with continuous potential and smooth wave functions with

convergent momentum, but with a potential bounded from below, although unbounded from above. Thus,

this system meets the first criterion stipulated above, but violates the other two. If we were to replace

criterion one with the requirement that V(x) be smooth, then the dynamically shifted oscillator as defined

currently would fail to meet this criterion as well. However, it is possible to replace the potential listed in

equation (1) with a smooth potential which gives identical numerical results to arbitrary accuracy.

4. Energy Eigenfunctions and Eigenvalues of the Dynamically Shifted Oscillator

Given that the algebraic and analytic techniques do not explain the dip in the energy eigenvalue, the

authors undertake a numerical study of eigenvalues and eigenfunctions for the dynamically shifted

oscillator.

In Fig. 2 we see 0 corresponding to different values of the reduced shift ξ0. Notice how the

eigenfunction splits into two peaks as ξ0 decreases from 1 to -3. An interesting effect we see is that the

corresponding eigenvalues behave as we might expect when ξ0=1, that is, ε0=1.4999. As we anticipate when

the potential energy is raised by one unit (over that of the harmonic oscillator), the energy eigenvalue rises

by one. However, when ξ0 decreases to -1 and -2 the energy eigenvalues drop below the standard ground

state solution of ε0=1/2. As we decrease ξ0 further to -3, we nearly recover the original ground state

eigenvalue of the harmonic oscillator.

Turning to Fig. 3, we see the evolution of the first excited state 1 as a function of the reduced dynamic

shift. Similar to the ground state, 1 appears to split into two separate waves as ξ0 becomes increasingly

negative. The figure, however, does not seem to show the corresponding eigenvalues hitting a minimum.

Notice the peculiarity happening with ε1 as ξ0 becomes increasingly more negative. Looking to Figure 6, we

see that ε0 and ε1 both approach 1/2 but ε0 from below and ε1 from above.

Evolutions of 2 and 3 as a function of the reduced dynamic shift ξ0 show similar behavior. As the

reduced dynamic shift decreases form 1 to -3, each eigenfunction splits into two branches. The values of ε2

and ε3 exhibit markedly similar behaviors to those of ε0 and ε1, respectively, but with both values now

approaching 1.5.

Turning our attention to Fig. 4, we begin to see degenerate behavior. When our dynamic shift is “large”

negative, in this case ξ0=-5, then we see ε0=0.5 and there are two eigenfunctions, one of even parity and one

of odd parity. However, both eigenfunctions are locally even about ±ξ0.

Fig. 5 shows the same situation for 1 and again there are two eigenfunctions, one of even parity and

one of odd parity, both corresponding to the same eigenvalue ε1=1.5. In this case, however, both

eigenfunctions are locally odd about ±ξ0.

Further investigation of the higher eigenfunctions and eigenvalues promulgate the evidence that the



dynamically shifted oscillator does indeed exhibit degenerate behavior. Following suit from Figures 4 and 5

we see the that eigenfunctions 2k are locally even and 2 1k are locally odd about ±ξ0.

Fig. 2. Ground-state eigenfunction of the dynamically shifted oscillator for different values of the dynamic

shift ξ0.

Fig. 3. The first excited state eigenfunction of the dynamically shifted oscillator for different values of the

dynamic shift ξ0.



Finally, Fig. 6 explicitly reveals how the eigenvalues split. The eigenvalues of the dynamically shifted

oscillator bifurcate with respect to the dynamic shift. The “upper” degenerate eigenvalues correspond to the

odd eigenfunctions, whereas the “lower” eigenvalues correspond to the even eigenfunctions.

Fig. 4. The even and odd ground state eigenfunctions of the dynamically shifted oscillator for a “large”

negative dynamic shift, in this case ξ0=-5. Note that both eigenfunctions are “locally” even.

Fig. 5. The even and odd first excited state eigenfunctions of the dynamically shifted oscillator for a “large”

negative dynamic shift, in this case ξ0=-5. Note that both eigenfunctions are “locally” odd.



Fig. 6. Bifurcation of energy eigenvalues of the dynamically shifted oscillator with respect to the dynamic

shift.

5. Conclusion

The dynamically shifted oscillator is an interesting quantum mechanical system in one dimension for

several reasons. Of primary importance is that it exhibits a degeneracy in eigenvalues in one dimension

when ξ0 3 . What's interesting about this is the fact that the standard algebraic techniques do not predict

any difference in energy from the standard harmonic oscillator. Moreover, the dynamic shift when negative

takes nondegenerate states and adiabatically moves them into degenerate states. One remaining question to

be resolved, since we have seen the numerical solution, is whether we have a true degeneracy or a

pseudo-degeneracy. In this case, the question becomes rather meaningless since the algebraic solution

suggests the all states are degenerate. Numerically, we can render this question irrelevant since we may in

fact have a separation of eigenvalues, but when ξ0 becomes large enough (negatively) then we will have

eigenvalues which are arbitrarily close in value. Furthermore, in a physical sense, the energy eigenvalues

certainly fall within the uncertainty limit and are therefore physically indistinguishable. In addition to this,

the potential can be made smooth by replacing the given potential on [ , ] with an arbitrarily high

degree (but finite) Fourier expansion which matches our value at V ( ) so as to guarantee smooth

solutions with degenerate eigenvalues. Finally, this system opens the door to finding multiple classes of

one-dimensional degenerate systems with continuous, symmetric potentials.

References

[1] Kar, S., & Parwani, R. R. (2007). Can degenerate bound states occur in one-dimensional quantum

mechanics? Europhysics Letters, 80(3), 30004.

[2] Koley, R., & Kar, S. (2007). Exact bound states in volcano potentials. Physics Letters A, 363, 369-373.

[3] Eisberg, R., & Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particle (2nd

ed.). New York: Wiley.

[4] Gettys, W. E., & Graben, H. W. (1975). Quantum solution for the biharmonic oscillator. American Journal

of Physics, 43, 626-629 .

[5] Janke, W., & Cheng, B. K. (1988). Statistical properties of a harmonic plus a delta-potential. Physics

Letters A, 129, 140-144.

[6] Hartmann, W. M. (1986). The dynamically shifted oscillator. American Journal of Physics, 54, 28-32.



[7] Mohazzabi, P. (2004). Theory and examples of intrinsically nonlinear oscillators. American Journal of

Physics, 72, 492-498.

Pirooz Mohazzabi received his PhD in materials science and engineering from

University of California, Berkeley, in1975. He joined University of Wisconsin-Parkside in

1986, where he is currently a professor of physics. He has published extensively in a

wide variety of areas, ranging from bicycle stability to cancer theory. His current

research focuses on thermodynamics and statistical mechanics of small systems, using

theoretical and computational methods including molecular dynamics and Monte Carlo

simulations.

Clark Alexander received his PhD in mathematics from Northwestern University in

2008. He is currently a visiting assistant professor at DePaul University in Chicago,

where he has been on faculty since 2011. His research interests are mathematical

physics, noncommutative geometry, and quantum algorithms.



Documents

Degeneracy in One Dimensional Quantum Systems: Dynamically